Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.2 → 0.8

Time: 3.7s

Precision: binary64

Math

\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]

↓

\[\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}} \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}}\right)\right)\right)\]

C

double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))));
}

↓

double code(double x, double y) {
	return ((double) (((double) cbrt(((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (x + 1.0)))))) * ((double) (x * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (((double) (x / y)) + 1.0)))) * ((double) cbrt(((double) (((double) (x / y)) + 1.0)))))))) * ((double) (((double) cbrt(((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (x + 1.0)))))) * ((double) cbrt(((double) (((double) cbrt(((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))))))))))))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	9.2
Target	0.1
Herbie	0.8

\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

1. Initial program 9.2

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 0.8

   \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)}\]

5. Applied associate-*r* 0.9

   \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.9

   \[\leadsto \left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{\color{blue}{1 \cdot \left(x + 1\right)}}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\]

8. Applied add-cube-cbrt 0.8

   \[\leadsto \left(x \cdot \left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}\right) \cdot \sqrt[3]{\frac{x}{y} + 1}}}{1 \cdot \left(x + 1\right)}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\]

9. Applied times-frac 0.8

   \[\leadsto \left(x \cdot \left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}}{1} \cdot \frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\]

10. Applied cbrt-prod 0.8

    \[\leadsto \left(x \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}}\right)} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\]

11. Applied associate-*l* 0.8

    \[\leadsto \left(x \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}}{1}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right)}\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\]

12. Simplified 0.8

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}}{1}} \cdot \color{blue}{\left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}}\right)}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\]

13. Final simplification 0.8

    \[\leadsto \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}} \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))